Le Monde puzzle [#920]

A pocket calculator with ten keys (0,1,…,9) starts with a random digit n between 0 and 9. A number on the screen can then be modified into another number by two rules:
1. pressing k changes the k-th digit v whenever it exists into (v+1)(v+2) where addition is modulo 10;
2. pressing 0k deletes the (k-1)th and (k+1)th digits if they both exist and are identical (otherwise nothing happens.
Which 9-digit numbers can always be produced whatever the initial digit?

I did not find an easy entry to this puzzle, in particular because it did not state what to do once 9 digits had been reached: would the extra digits disappear? But then, those to the left or to the right? The description also fails to explain how to handle n=000 000 004 versus n=4.

Instead, I tried to look at the numbers with less than 7 digits that could appear, using some extra rules of my own like preventing numbers with more than 9 digits. Rules which resulted in a sure stopping rule when applying both rules above at random:

code that fills an occupancy table for the numbers less than a million over 10⁶ iterations. The solution as shown below (with the number of zero entries over each column) is rather surprising in that it shows an occupancy that is quite regular over a grid. While it does not answer the original question…